Connection to TTSA [1]}: 
When \(n=1\) , Algorithm 1 reduces to an algorithm where at each step, \(y_{k+1} = y_k - \beta h_g^{k}\)  and \(x_{k+1} = \mathrm {proj}_{\mathcal {X}}(x_k-\alpha h_x^k)\) . The \(\lambda \)  updates go away because \(\lambda _k = 1\)  for all \(k\) . This algorithm, then, is exactly the TTSA algorithm. In the multi-task case where \(n>1\) , our algorithm is a natural generalization of TTSA, with \(y\)  updates behaving identically and \(\lambda \)  updates inspired by gradient descent-ascent algorithms such as [2]}. Our framework expands upon TTSA by allowing the optimization of multiple different functions \(g_i\)  at once.
